Polarization Based Out-Coupling for Cavity Based X-ray FELs

Cavity Based X-ray Free-Electron Lasers (CBXFELs) promise fully 3D coherent, very brilliant and shot-to-shot stable X-ray pulses. For CBXFELs the X-ray radiation is trapped in an X-ray optical cavity, which is formed using Bragg-reflecting crystal mirrors. As is the case for laser systems in the optical regime, a major question for the CBXFELs is how to couple out the radiation from the cavity. Possibilities range from employing semi-transparent thin crystals, over manipulation of the electron phase space density to cavity dumping schemes. In this work, making use of the strong polarization of Bragg reflection shall be studied for out-coupling. As the radiation does not change its polarization during Bragg reflection, reflection in the direction of the polarization vector is suppressed. By adjusting the 3D orientation of the crystals with respect to the polarization axis of a linearly polarized undulator, or vice versa, the transmission through the crystals can be tailored to some degree independently of the crystal thickness.


Introduction
Due to the lack of monochromatic external seeding sources in the hard X-ray regime, current hard X-ray Free-Electron Laser (XFEL) machines such as the Linac Coherent Light Source (LCLS), the European XFEL (EuXFEL), the Spring-8 Angstrom Compact free-electron LAser (SACLA), the SwissFEL and the Pohang Accelerator Laboratory X-ray Free Electron Laser (PAL-XFEL) are mainly using the seedless self-amplified spontaneous emission (SASE) scheme for operation.These sources produce very brilliant femtosecond X-ray pulses with a high degree of transverse coherence.Being initiated from the statistical noise of spontaneous emission, they do, however, suffer from a low degree of longitudinal coherence, with a typical radiation pulse of some 10femtoseconds length, consisting of thousands of longitudinal modes, as well as a strong shot-to-shot fluctuations.A promising class of schemes still to be realized are the Cavity Based X-ray FELs.This term includes both the high-gain X-ray Regenerative Amplifier FEL (XRAFEL) proposed by Z. Huang in 2006 [1] as well as the low-gain X-ray Free Electron Laser Oscillator (XFELO) proposed by K.J. Kim in 2008 [2].The fundamental commonality of these schemes are that they are based on trapping FEL radiation inside a X-ray optical cavity, using monochromatizing crystals based on Bragg reflection instead of total reflecting optical mirrors.Due to the promise of delivering outstanding radiation properties, CBXFELs have received growing interest in the recent years [3,4,5,6,7,8,9,10,11,12,13,14].Currently, there are two projects to install proof-of-concepts CBXFEL experiments at the European XFEL [15] and the Linac Coherent Light Source-II (LCLS-II) [16], respectively.Also, at the still to be completed Shanghai High-Repetition-Rate XFEL and Extreme Light Facility (SHINE) a full cavity enhanced FEL beamline is planned [17].
Same as for optical laser cavities, a major challenge of CBXFELs is how to efficiently couple out the radiation.The options to do that range from X-ray optics based techniques (see for example [18,19,13,12]) to electron beam based ones (see for example [1,20,21,14]).In this contribution, the use of the strong polarization dependence of Bragg reflection is studied as an X-ray optics based approach for radiation out-coupling, applicable both to low-gain XFELO sources, requiring low cavity losses, as for high-gain XRAFEL sources, which can take advantage of high out-coupling rates.

Polarization dependence of dynamic diffraction
For the calculation of the reflection efficiency at the crystals, dynamic diffraction theory is applied throughout this contribution.In order to keep the algebra as simple as possible, a couple of approximations are applied, which are well justified for the perspective crystal used in CBXFELs.First, the two beam approximation is applied, meaning that the reflection condition is only met for one single set H of parallel crystal planes.Second, the angle of incidence is assumed to be much bigger than 0, neglecting grazing incidence.Third, only symmetric Bragg (backward) reflection is considered, meaning that the surface of the crystal and the reflecting planes are the same, which cancels possible dispersion.And lastly, the crystal itself is assumed to be perfect and not-strained, which strongly simplifies the algebra1 .Generally, the system of equations for the dynamic diffraction is dependent on the polarization.A convenient orthornormal basis set for polarization of the incoming beam with wavevector K 0 and for the reflected beam with wavevector This choice of basis perfectly decouples the polarization components σ and π from each other.Then, the reflectivity r σ,π and transmissivity t σ,π for a crystal of thickness t c at wavelength λ and, hence, wavenumber k = | K 0 | = 2π/λ, and Bragg angle Θ B = arcsin (− K0 H/k| H|) become [22] In above equations, χ 0,H is the Fourier component of the material dependent susceptibility.P σ,π a polarization dependent factor, which is P σ = σ0 σH = 1 and P π = π0 πH = sin (2Θ B ).The strong dependence of P π on the Bragg angle Θ has a strong influence on the reflectivity and transmissivity and is the essential quantity for this study.This is emphasized in Fig. 1 14th  For the following, we assume a linearly polarized incoming radiation pulse E in (r, λ) = E in (r, λ)x from planar undulators in spectral domain, where without loss of generality the polarization is assumed in x-direction.Also, it is approximated that the cavity is, other than the crystals, only made up out of linear, lossless elements and that the reflection is independent of the higher order transverse moments of the pulse like divergence and width.Then it is sufficient for this study to look at the spectral components S(λ).λ will be omitted in the following.For a four crystal cavity of constant Bragg angle are the radiation reentering the undulator as seed and the radiation transmitted through the first undulator, respectively.Above relations show, that not only the magnitude of reflection and transmission are effected by the polarization, but also the polarization of the seed and of the transmitted pulse can be rotated.While for the transmission this should not matter for most experiments, for the seeding radiation the y-component does not couple to the FEL process in the planar undulator and does, thus, not contribute to the seeding.Hence, we term it "lost" in the following.
In order to be more quantitative, the coordinate system of the undulators needs to be set into relation with the crystal coordinate system.For the undulator system, the downstream propagation direction of the radiation is set as ẑU and the polarization as xU .For the crystal coordinate system, the vector H is defined as parallel to −ẑ C and the xC and ŷC coordinates can be chosen as random orthonormal vectors.The coordinate transformation can be set by forms of a rotation matrix with two rotational degrees of freedom around the xC and ŷC axis2 , respectively called pitch and roll in the following, equivalent to a mechanical rotation setup.
Then the transformation matrix from the crystal to the undulator frame becomes with a σ,π = r 4 σ,π for refl and t σ,π for tr.In the top row, a full angular scan over both pitch and roll is shown, while in the bottom row the angles are moved along one of the lines depicted in the top row, keeping the Bragg angle fixed to a specific value.The calculations were done for a comparably thick t c =100 µm diamond with <1 0 0> surface orientation3 , which is the crystal of choice for the European XFEL CBXFEL experiment [20].From the transmission plot in the top right as well as the green lines in the bottom row, we see that the out-coupling technique is very sensitive to the Bragg angle, in accordance with the dependence of the polarization factor P π on Θ B , with the transmission quickly dropping when moving away from Θ B = 45 • .An interesting observation from the bottom row plots is, in accordance with Eq. ( 5), that when varying the angles along a constant Θ B line, not only the transmission fraction is effected but also the polarization direction.The polarization loss plot in the top middle as well as the constant Θ B = 45 • plot in the bottom left show a major downside of the technique.In the cases of high transmission, also the polarization loss becomes very high, reaching fractions higher than S seed ŷU /|S in | > 20 %.Especially for a rectangular cavity with Θ B = 45 • , when going for transmission |S tr |/|S in | ≥ 40 %, the loss fraction actually becomes bigger then the seeding fraction.This makes the polarization based out-coupling approach rather ineffective for XRAFEL type applications, where high out-coupling efficiency is ideal to cope with the high gain.It might still be interesting for demonstrator type setups due to its simplicity.For low-gain XFELO type setups which require out-coupling efficiencies on the order of 5 %, though, the out-coupling technique is appealing.For the Θ B = 45 • case, the seeding loss at |S tr |/|S in | = 5 % still amounts to |S seed |ŷ U /|S in | ≈ 5 % and the total losses, also including transmission and absorption losses, is 1 − |S seed |x U /|S in | ≈ 12 %, which is quite much, but might be tolerable.For Θ B s moving away from the 45 • line, the fraction of polarization loss actually decreases.However, this is due to an increase of reflectivity r π , which is having a much larger penetration depth and, hence, more absorption.Thus, the actual total loss fraction of seeding radiation is increasing with increasing Θ B .This can also be seen from Fig. 3.The figure also shows that the losses are strongly dependent on the crystal thickness, effecting the total absorption, when moving away from the rectangular type cavities at Θ B = 45 • for which r π = 0. Also, by lowering the desired transmission, the loss fraction can also be decreased.

Discussion and Outlook
Above analysis demonstrates that while the polarization based out-coupling can yield very high transmission fraction even with 'thick' crystals, it comes with the major downside of significant additional losses.It is clear that for the technique to be effective, it is important to stay very close or ideally on the Θ B = 45 • line.Nonetheless, compared to other out-coupling techniques of similar simplicity, in particular the sideband transmission initially used for the EuXFEL experiment [20] or thin drum head crystals, this downside can still be preferable to the decreased time-bandwidth product of the prior and the reduced mechanical stability of the latter method.As such, it might still be interesting for the rectangular Θ B = 45 • CBXFEL demonstrator at SLAC.Furthermore, it is possible to diminish the polarization loss problem by using hard X-ray crystal phase plates, for example made from diamond, as they are common for synchrotron sources [23].Using one phase retarder at the end of the undulators, in order to more easily manipulate the incoming polarization, and one retarder at the beginning, to turn the polarization back to obtain full seeding, the polarization based out-coupling could be majorly improved.This could make the technique also applicable to XRAFEL type sources.Another possibility to influence the polarization state without the necessity of tuning the crystal is the usage of variable polarization undulators (see for example [24]).This approach does however suffer from the same drawbacks as analyzed in the sections above.
Another small modification to adapt the concept to XRAFELs is to use a tapered afterburner with the polarization axes shifted by 90 • .A similar approach has also been studied by N. Huang et al. [25] for polarization control of XFELOs without concern for the out-coupling.For a rectangular cavity, the crystals could be aligned such that he first x-axis polarized undulator cells would be reflected by nearly 100 %, while the y-axis polarized afterburner would be fully transmitted.By tuning the efficiency of the afterburner by a careful non-linear taper, as discussed in [25], one could tune the out-coupling rate and principally go to very high absolute transmission.However, as high pulse energies are being transmitted through the entire crystal and being absorbed on the way, high amounts of total heat are being deposited.This could strongly destabilize the CBXFEL source [20] and requires further studies.

Figure 2 .
Figure 2. The top row shows the seeding efficiency S seed xU /|S in |, the polarization loss S seed ŷU /|S in | and the transmission |S tr |/|S in |.The red, green, and purple lines in the top row display angular combinations with constant Θ B .These lines correspond to the plots in the bottom row, where the roll is varied along with the pitch to yield cos(R) = sin(P )/ sin(Θ B ).On top of the green transmission line in the bottom plot the actual polarization directions of the transmitted radiation are sketched.

Figure 2
Figure 2 displays the effective seeding radiation S seed xU /|S in |, 'lost' radiation S seed ŷU /|S in | and transmitted radiation |S tr |/|S in | for a scan of angles.In the top row, a full angular scan over both pitch and roll is shown, while in the bottom row the angles are moved along one of the lines depicted in the top row, keeping the Bragg angle fixed to a specific value.The calculations were done for a comparably thick t c =100 µm diamond with <1 0 0> surface orientation 3 , which is the crystal of choice for the European XFEL CBXFEL experiment[20].From the transmission plot in the top right as well as the green lines in the bottom row, we see that the out-coupling technique is very sensitive to the Bragg angle, in accordance with the dependence

Figure 3 .
Figure 3. Seeding efficiencies |S seed |x U /|S in | ≈ 5 % against Bragg angle Θ B for various diamond thicknesses at two different desired transmission fractions.The end of a line signifies that the desired transmission cannot be achieved.